The order of attention allocations in visual search is often explained by the operation of a pre-attentive priority map. If the search target is salient, thus achieving a high weight on that map, it typically summons attention immediately (efficient search). However, if a task-irrelevant item receives an even higher weight, attention is captured by this 'distractor' before it can be reallocated to the target. Although attentional capture is a widely examined phenomenon, it is still unclear what processes are required for continuing search following a misallocation of attention: does attention simply move on to the next item, or must the attended distractor be actively suppressed on the priority map? To examine this, we devised a task in which attention was reliably allocated first to the distractor and only then to the search target, as evidenced by an electrophysiological marker of attention allocation (N2pc). Importantly, attention allocation to the distractor was reliably followed by an electrophysiological correlate of active suppression (PD). Given this, we asked whether this suppression works instantaneously, independently of the distractor weight on the priority map, or whether it takes time dependent on that weight. We contrasted these two possibilities in computational models that either did, or did not, include a parameter reflecting distractor-weight-dependent time requirements for distractor suppression. Only model variants that included such a parameter could successfully explain the observed RT data patterns. Together, these findings indicate that distractor suppression is necessary for continuing search after attention was captured and that the time required for this process depends on the distractor weight on the priority map. We propose that the same mechanisms are involved when attention is misallocated towards non-salient distractors during inefficient search and that persistent suppression might act as a transient memory of visited distractor locations.